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1 On quantum phase crossovers in finite systems

Clare Dunning, Katrina Hibberd, Jon Links

Published Jan 1, 2008• 6 citations
OpenAlex

Abstract

In this work we define a formal notion of a quantum phase crossover for certain Bethe ansatz solvable models. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-body Schrödinger operator. Bifurcations of the minima for the potential of the Schrödinger operator determine the crossover couplings. By considering the behaviour of particular ground-state correlation functions, these may be identified as quantum phase crossovers in the many-body integrable system with finite particle number. In this approach the existence of the quantum phase crossover is not dependent on the existence of a thermodynamic limit, rendering applications to finite systems feasible. We study two examples of bosonic Hamiltonians which admit second-order crossovers. Keywords: quantum integrability (Bethe ansatz), Bose-Einstein condensation (theory) 1

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In this work we define a formal notion of a quantum phase crossover for certain Bethe ansatz solvable models. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-body Schrödinger operator. Bifurcations of the minima for the potential of the Schrödinger operator determine the crossover couplings. By considering the behaviour of particular ground-state correlation functions, these may be identified as quantum phase crossovers in the many-body integrable system with finite particle number. In this approach the existence of the quantum phase crossover is not dependent on the existence of a thermodynamic limit, rendering applications to finite systems feasible. We study two examples of bosonic Hamiltonians which admit second-order crossovers. Keywords: quantum integrability (Bethe ansatz), Bose-Einstein condensation (theory) 1
In this work we define a formal notion of a quantum phase crossover for certain Bethe ansatz solvable models. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-body Schrödinger operator. Bifurcations of the minima for the potential of the Schrödinger operator determine the crossover couplings. By considering the behaviour of particular ground state correlation functions, these may be identified as quantum phase crossovers in the many-body integrable system with finite particle number. In this approach the existence of the quantum phase crossover is not dependent on the existence of a thermodynamic limit, rendering applications to finite systems feasible. We study two examples of bosonic Hamiltonians which admit second-order crossovers.
The aim of this work is to develop a technique for identifying quantum phase transitions which does not rely on the existence of a thermodynamic limit, for studies in finite systems. The approach we adopt exploits an exact mapping of the spectrum of a many-body integrable system, which admits an exact Bethe ansatz solution, into the quasi-exactly solvable spectrum of a one-particle Schrodinger operator. Bifurcations of the minima for the potential of the Schrodinger operator determine critical ground-state couplings. By considering the behaviour of certain ground-state correlation functions, these may be identified as quantum phase transitions in the many-body integrable system with finite particle number. We study two particular examples of bosonic Hamiltonians which admit second-order transitions, and discuss further applications.
We introduce a variational approach for the quantum inverse scattering method to exactly solve a class of Hamiltonians via Bethe ansatz methods. We undertake this in a manner which does not rely on any prior knowledge of integrability through the existence of a set of conserved operators. The procedure is conducted in the framework of Hamiltonians describing the crossover between the low-temperature phenomena of superconductivity, in the Bardeen–Cooper–Schrieffer theory, and Bose–Einstein condensation. The Hamiltonians considered describe systems with interacting Cooper pairs and a bosonic degree of freedom. We obtain general exact solvability requirements which include seven subcases that have previously appeared in the literature.
Introduction The one-dimensional Bose gas with point-like interaction of the particles (the quantum variant of the nonlinear Schrödinger equation) is one of the main and most important models which can be Solved by means of the Bethe Ansatz. This model has been thoroughly investigated (and. We shall start with the construction of eigenfunctions of the Hamiltonian in a finite volume. Quantities interesting from the physical point of view (in the thermodynamic limit at zero temperature) are considered; the thermodynamics at finite temperatures is investigated in detail as well. A number of essential ideas which will be applied to other models are introduced. The construction of eigenfunctions of the Hamiltonian is explained in section 1. Their explicit form and, in particular, the two-particle reducibility, are common features of the models solvable by means of the Bethe Ansatz method. Periodic boundary conditions are imposed on the wave function in section 2; the Bethe equations for the particle' momenta are introduced and analyzed. Taken in the logarithmic form, these equations realize the extremum condition of a certain functional, the corresponding action being called the Yang-Yang action. The transition to the thermodynamic limit is considered in section 3. In that same section the ground state of the gas is constructed. The density of the particle distribution in momentum space and the energy of the ground state are calculated. The method of the transition to the thermodynamic limit described in this section is rather general and may be applied to any model solvable by means of the Bethe Ansatz. In section 4, excitations above the ground state are constructed and their main characteristics (energy, momentum and scattering matrix) are determined by means of the dressing equations. The ground state of the model is the Dirac sea (also called the Fermi sphere).
We demonstrate a novel approach that allows the determination of very general classes of exactly solvable Hamiltonians via Bethe ansatz methods. This approach combines aspects of both the co-ordinate Bethe ansatz and algebraic Bethe ansatz. The eigenfunctions are formulated as factorisable operators acting on a suitable reference state. Yet, we require no prior knowledge of transfer matrices or conserved operators. By taking a variational form for the Hamiltonian and eigenstates we obtain general exact solvability conditions. The procedure is conducted in the framework of Hamiltonians describing the crossover between the low-temperature phenomena of superconductivity, in the Bardeen-Cooper-Schrieffer (BCS) theory, and Bose-Einstein condensation (BEC).
Exactly solvable models provide a unique method, via qualitative changes in the distribution of the ground-state roots of the Bethe ansatz equations, to identify quantum phase transitions. Here we expand on this approach, in a quantitative manner, for two models of Bose–Einstein condensates. The first model deals with the interconversion of bosonic atoms and molecules. The second is the two-site Bose–Hubbard model, widely used to describe tunneling phenomena in Bose–Einstein condensates. For these systems we calculate the ground-state root density. This facilitates the determination of analytic forms for the ground-state energy, and associated correlation functions through the Hellmann–Feynman theorem. These calculations provide a clear identification of the quantum phase transition in each model. For the first model we obtain an expression for the molecular fraction expectation value. For the two-site Bose–Hubbard model we find that there is a simple characterization of condensate fragmentation.